Magnetic Neutron Scattering - Appunti

X

−iq·r n ' − ·

e = (2n + 1)(−i) j (ρ)P (cos θ) j (ρ) iq r[j (ρ) + j (ρ)] (2.1)

n n 0 0 2

n=0

with the truncation valid for small values of ρ. Introducing this expansion in the ex-

pression for Q , we find

p 1 0 0 0 0

× × ×

[j (ρ) + j (ρ)]q̂ l q̂ + Q = r p

Q = }l

p 0 2 p

2

where we have introduced

i 1

0 0 0 0

× · ·

Q = q̂ j (ρ)p + [j (ρ) + j (ρ)][(q̂ r)p + (q̂ p )r] + ...

0 0 2

p 2}

}q

H

If is considered to be the Hamiltonian for the electron, then

e dr i

0

p = p + A = m = m [H, r]

e

c dt }

9

10 CHAPTER 2. MAGNETIC SCATTERING BY A CRYSTAL

0 may be written

and Q p

m iq

0 ·

× −j

Q = [j (ρ) + j (ρ)][H, (q̂ r)r] + ...

q̂ (ρ)[H, r] + 0 2

0

p 2 q 2

}

Considering an arbitrary operator Â, we have

hi|[H, i − i

Â]|f = (E E )hi|

Â|f

i f 0

→

from which we see that, in the limit q 0, the Q vector doesn’t contribute to the

p

6

cross section. At q = 0 we cannot exploit the same result because j (ρ) does not

n

H.

commute anymore with However, if we restrict ourselves to scattering processes in

which the l quantum number is conserved (∆l = 0 selection rule), the matrix element

H

coming coming from the first term vanishes identically, because j (ρ) and are both

0 H

diagonal with respect to l. The second term, instead, can be simplified by replacing

with the kinetic-energy operator. Through some algebra, we obtain that it’s composed

of a radial times an angular operator

0 ' × ·

Q (q̂ r̂)(q̂ r̂)Q

r

p

with i

† 2 2 2 2

− [j (ρ) + j (ρ)][∇ , r ] + [∇ , r ][j (ρ) + j (ρ)]

Q = Q = 0 2 0 2

r r 8

Our next assumption is that the radial part of the wave function, as specified by the

principal quantum number n, and by l, is the same in the initial and the final state,

hi|Q |f i hnl|Q |nli

i.e., both n and l are unchanged. In this case, = = 0, since Q is

r r r

an imaginary Hermitian operator.

|ii |f i |(nls)m i,

The assumption that and are linear combinations of the states m

l s

where (nls) is constant, implies that the radial and angular dependencies are factor-

ized, both in the expansion of the operators and in the wave functions. Introducing

the vector operator K(q), defined so that

hi|q̂ × × i hi|Q |f i

K q̂|f = + Q

p s

we can write its orbital part following the assumptions made till now

1 hj

K (q) = [hj (q)i + (q)i] l

p 0 2

2

with

2.1. ELASTIC AND INELASTIC MAGNETIC SCATTERING 11

∞

Z 2 2 ·

hj r R (r)j (q r)dr

(q)i = n

n 0

where R(r) is the normalized radial wave function. The assumption that the final and

initial states have the same parity implies that only the terms in the expansion (2.1)

for which n is odd may contribute to K . By the same argument, the spin part only

p

involves terms with n even. The following result obtained for K(q) is the basis for the

dipole approximation for the scattering cross-section

1 1

hj hj

K(q) = (q)i(l + 2s) + (q)il (2.2)

0 2

2 2

Within this approximation, it is straightforwardly generalized to the case of more than

one electron per atom, as the contributions are additive, in the sense that l and s

2

P P

are replaced by L = l and S = s, and R (r) by the normalized distribution for

all unpaired electrons belonging to the atom at R . Therefore, the total interacting

j

Hamiltonian ca be written as X

H · × ×

(q) = 8πµ µ (q̂ K (q) q̂) (2.3)

int B n j

j

from which the total magnetic cross-section is obtained

2

0

2 2

∂ σ k e

}γ X X

X

n − P (β)× (2.4)

= (δ q̂ q̂ ) i

αβ α β

2

∂E∂Ω k mc 0 i,f

αβ jj

−iq·R iq·R

×hi|K |f ihf |K |iiδ(}ω −

0

(q)e (q)e + E E )

j j

0

jα j β i f

2.1 Elastic and Inelastic Magnetic Scattering

Elastic scattering is the characteristic defining structure of a solid. Apart from the

trivial q = 0 forward-scattering process, it is absent in a liquid or a gas. Using an

adiabatic approximation, we can separate, in the differential cross-section, the average

over the ions and electrons states, obtaining the thermal average

(0) (0)

−iq·(R −R )

−iq·(R −R −2W

) (q)

he i

0 0

= e e

j j

j j

(0)

where we have exploited the fluctuations R = R + u , which, due to the harmonic

j j

j

12 CHAPTER 2. MAGNETIC SCATTERING BY A CRYSTAL

approximation, give rise to the Debye-Waller factor. The expansion of the exponen-

hq · i,

tial has been truncated at the zero order in u neglecting contributions from the

j

inelastic phonon scattering. This allows us to write ∞

Z

1

X iωt hK

− dte (q, t)K (q, 0)i

P (β)hi|K (q)K (q)|iiδ(}ω + E E ) = 0

0 jα j β

i jα j β i f 2π} −∞

i

where we have used the integral representation of the δ-function, and K(t) is the op-

erator in the Heisenberg picture. Thus, the cross-section reduces to

2

0

2 2

∂ σ k e

}γ X

n −2W (q) −

= N e (δ q̂ q̂ )S (q, ω)

αβ α β αβ

2

∂E∂Ω k mc αβ

where we have introduced the Van Hove scattering function

∞

Z

1 (0) (0)

−iq·(R −R

X )

iωt hK

0

dte e (q, t)K (q, 0)i

S (q, ω) = j j 0

jα j β

αβ 2π}N −∞ 0

jj

hK ihK i

If is added and subtracted, we can distinguish two contributions, a static

0

jα j β

(self) and a dynamic (distinct) one αβ

αβ αβ

S (q, ω) = S (q) + S (q, ω) (2.5)

s d

where the static or elastic component is

1 (0) (0)

−iq·(R −R

X )

αβ hK 0

(q)ihK (q)ie

S (q) = δ(}ω) j j

0

jα j β

s N 0

jj

while the inelastic one can be related to the magnetic susceptibility through the

fluctuation-dissipation theorem 1 1

αβ 00

S (q, ω) = χ (q, ω)

αβ

d −β}ω

−

π 1 e

The above relation is extremely useful in analysing neutron magnetic scattering. First,

it establishes a direct way of comparing the results with those of absorption mea-

00

surements, such as electron spin resonance (ESR) that probe χ directly [2]. Second,

dynamical susceptibility often is the quantity that arises in theoretical calculations,

for example, in the random phase approximation (RPA). Third, dynamical suscepti-

bility often can be described by very simple physical models, for instance, a damped

harmonic oscillator.

2.1. ELASTIC AND INELASTIC MAGNETIC SCATTERING 13

2.1.1 Magnetic Bragg Scattering

If the magnetic moments in a Bravais lattice are ordered in a static structure, described

by the wave vector Q, we may write

1

(0)

(0) −iQ·R

∗

iQ·R hK

hK

hK + (q)i e

(q)ie

(q)i = j

j α

α

jα 2

hK i

allowing to be complex in order to account for the phase. The static contribution

α

is then proportional to 3

(2π) 1

X

∗

αβ <{hK }δ(}ω) − − −

S (q) = (q)ihK (q)i (1+ δ ){δ(τ + Q q)+ δ(τ Q q)}

α β Q,0

s v 4

τ

where δ is equal to 1 in the ferromagnetic case Q = 0, and zero otherwise, and v and

Q,0

τ are respectively the volume of a unit cell and a reciprocal lattice vector. We see that

this magnetic order leads to magnetic Bragg scattering whenever the scattering vector

±Q

is equal to plus a reciprocal lattice vector. This contribution to the cross-section

is the one measured in neutron diffraction experiments, in which all neutrons in

0

the direction k are observed without the use of the analyser crystal as an energy dis-

crimination. This consequence is highlighted by the fact that the static contribution

is overwhelmingly the most intense.

For a crystal with a basis of p magnetic atoms per unit cell, the ordering of the mo-

ments corresponds to

1 (0)

∗ −iQ·R

iQ·R

hK hK hK

(q)i = (q)ie + (q)i e R = R + d

js js

jsα sα sα js s

j

2

with s = 1, 2, ..., p, and d the vector determining the equilibrium position of the s-th

s

atom in the unit cell. This has the effect of introducing a geometric form factor in the

summation over the atoms

p p

N (0) (0)

X X X X

−iq·(R −R

−iq·(R −R −iq·d

)

) 2

|F

e = e (q)| ; F (q) = e

is jr s

i j G G

s,r s=1

ij ij

such that 2

2

dσ e

}γ X

n −2W (q) −

= N e (δ q̂ q̂ )|hK (q)ihK (q)i|×

0 αβ α β α β

2

dΩ mc αβ

3

(2π) 1

X ∗

× − − −

(1 + δ )<{F (τ )F (τ )}{δ(τ + Q q) + δ(τ Q q)} (2.6)

Q,0 α β

v 4

τ

14 CHAPTER 2. MAGNETIC SCATTERING BY A CRYSTAL

where N is the number of unit cells, and the form factor is

0 τ

X −iτ ·d

hK

F (τ ) = (q)ie s

α sα

s=1

which can make some magnetic Bragg peaks disappear due to destructive interference.

Equation (2.6) applies in the most general case of magnetic ordering with a single wave

vector Q. It describes well a variety of particular situations, such as flat spirals, longi-

tudinal spin-density waves, helimagnets, anti-ferromagnets (e.g. for Q = (π, π, π)) and

a ferromagnets. The fundamental reason behind the importance of single-Q modulated

structures is that only such states arise in a single second order phase transition [2].

2.2 Magnons

Thermal energy and quantum zero-point fluctuations cause the relative orientation

of individual magnetic moments in an ordered structure to fluctuate. Because the

spins are coupled to one another by exchange interactions, the normal modes of these

fluctuations are collective